Questions on the mathematics behind the famed toy invented by Ernő Rubik.

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Number of unique permutations of a 3x3x3 cube

Given a 3x3x3 cube (like a rubik's cube) where each of the 27 cubes has a distinct number, how many unique permutations are possible? Simple rotations of the entire cube should not be counted. The ...
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0answers
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identifying a subgroup of $S_8$ generated by 4-cycles

Let $G \subseteq S_8$ be the subgroup generated by some 4-cycles. If we number the elements $1,2,\dots, 8$, the 4-cycles are $(1234),(5678),(1485),(2376),(1265),(4378)$ I am not sure if I have ...
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1answer
123 views

Creating a Rubik's Cube Algorithm

So, I realize this isn't a very math-oriented question, but I thought it fit better here than anywhere else. Anyway, I was simply wondering how people come up with the many Rubik's Cube algorithms, ...
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27 views

Rubik's cube with different neighboring color

How many valid configuration of the 3x3 Rubik's cube are there so that no two adjacent faces are the same color? Thanks.
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3answers
108 views

Minimum number of random moves needed to uniformly scramble a Rubik's cube?

Follow up from my last question: $3 \times 3$ Rubik's cube scrambling question I am talking about $3 \times 3$ Rubik's cubes. Start with a solved cube. Then make some amount of random moves ...
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$3 \times 3$ Rubik's cube scrambling question

It was proven that a Rubik's cube needs at most $20$ moves to solve. This implies that any configuration of a Rubik's cube can be reached from an unscrambled Cube in at most $20$ moves. So, say when ...
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1answer
34 views

Eulerian path for Rubik's Cube states

There are a number of discussions online confirming that there exists a Hamiltonian cycle through the states of a Rubik's Cube. Or more precisely, the "quarter-turn metric Cayley graph for the Rubik's ...
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134 views

Can RUBIK's cube be solved using group theory?

Can RUBIK's cube be solved using group theory? If yes, how can we use it to solve a $2\times2$ Rubiks Cube?
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1answer
49 views

Independently rotating cubelets of a 2 x 2 rubiks cube?

I was reading up on 2 x 2 Rubik's cube. Wikipedia says that Any permutation of the eight corners is possible (8! positions), and seven of them can be independently rotated (37 positions). I ...
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1answer
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What is “tetrad twist” in relation to Rubik's cubes?

I am trying to write a computer program to solve Rubik's cubes using a version of Thistlethwaite's algorithm. Instead of using lookup tables to perform each phase of the algorithm, I'm using a ...
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3answers
3k views

Can a Rubik's cube be mapped knowing only two sides?

Is it possible to know the entire configuration of a Rubik's cube looking at only two sides and not rotating the cube? In other words: what is the minimum information required to create a ...
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1answer
51 views

What mathematical algebra explains sequence of circular shifts on rows and columns of a matrix?

Consider a simple matrix (3X3) with entries thus: [1 2 3; 4 5 6; 7 8 9;] Circular shifts can be performed on any row or any column thus: row-(1/2/3)-(right/left) and column-(1/2/3)-(up/dn) ...
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52 views

Rubik cube number of alternative solutions

If a cube is in a configuration that requires 20 moves to solve, is that sequence unique, or are there multiple sequences that arrive at a solution? That is: are there are two or more sequences that ...
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2answers
276 views

Is it possible to shuffle a 3x3 Rubik's cube so that there's no more than 2 pieces of the same color in every face?

I'm not sure if this question belongs here but I see lots of Rubik Cube's questions around so here it goes: Can I take a standard 3x3 Rubik's Cube and shuffle it so that, for every face, there are no ...
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4answers
201 views

Unreachable rubik cube positions.

Are there positions of the rubik cube which cannot be reached by applying the standard moves starting from the solved cube? If so, how many such positions are there?
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Rubik's cube question showing surjectivity

Suppose $f: G \rightarrow S_C$ be the action of Rubik's group G on the collection $C$ of corner cubelets. I want to show that the action $f$ is onto $S_C$.
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Rubik's Group Question - Commutators

Suppose that $f: G \rightarrow S_C$ be the action of Rubik's group $G$ on the collection of $C$ of corner cubelets. So if $C_1$ and $C_2$ be distinct corner cubelets. Show that there is a move $M \in ...
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2answers
107 views

What are the total number of distinguishable configurations of a rubik's cube?

Given a standard $3 \times 3 \times 3$ rubik's cube, what are the total number of distinguishable configurations of the cube? Two configurations are called distinguishable iff one can be rotated to ...
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0answers
50 views

How to “validate” a Rubik's Cube configuration? [duplicate]

Looking at a picture of a Rubik's cube how can I tell if it's in one of its possible configurations (image is not fake)? By "possible configuration" I mean a configuration that can be achieved by ...
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0answers
136 views

Faces on a rotating cube

This one is probably a softball question, but... Fact: A cube has 6 faces Fact (unless my math is wrong): A cube that is cut in half center horizontal can be reoriented to have 18 unique faces. ...
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2answers
320 views

combinations of Rubik cube

A Rubick's Cube has owl heads on it, which can be misoriented. How many (times) MORE combinations are there of this cube vs. one that has blank stickers? Can someone give me some hints? Thanks
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1answer
251 views

Rubik's cube and counting

Inspired by this question, I formulate the following: Suppose I have a $3\times3\times3$ Rubik's cube, call each small square on the surface a piece, there are then $3*3*6 = 54$ pieces. Enumerate ...
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383 views

Literature on group theory of Rubik's Cube

While searching for literature on the group theory of Rubik's Cube, I mostly find introductions to group theory motivated by applications to Rubik's cube. I.e. the focus lies on elementary group ...
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Why should Rubik's cube get attention from mathematicians?

I've seen a lot of math debate, calculations and other stuff related to Rubik's cube lately, but I don't really understand why is it important, why should anyone spend time asking and answering ...
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Rubik Cube finite non-abelian group

I was reading the following paper from MIT: http://web.mit.edu/sp.268/www/rubik.pdf The paper is not difficult to understand, it is more or less a short introduction into group theory, taking the ...
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150 views

Rubik's cube as $2\times2\times2$ or $2\times2$? [closed]

I'm surprised why people call a Rubik's cube as $2 \times2\times2$ or $3\times3\times3$? It's a CUBE! Isn't saying $2\times2$ or $3\times3$ cube (or rather just $2$ or $3$) sufficient?
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Lower bounds on numbers of arrangements of a Rubik's cube

Last night, a friend of mine informed me that there were forty-three quintillion positions that a Rubik's Cube could be in and asked me how many there were for my Professor's Cube (5x5x5). So I gave ...
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4answers
2k views

Minimum number of steps needed to solve a rubik cube

Long time ago I've seen a book on group theory and there was an appendix about rubik cube. I remember there were only three steps that enabled me to solve my cube (three strings with letters encoding ...
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1answer
122 views

The redundancy of Rubik's cube states [duplicate]

Possible Duplicate: Rubik’s Cube Not a Group? I take a Rubik's cube in the solved state, and I secretly assign a unique integer label to each of the cubies. I then, via an arbitrarily long ...
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2answers
356 views

Rubik's Cube Combination

Could anyone explain why the number of legal or reachable combinations of a $3\times 3\times 3$ Rubik's Cube is $1/12\mbox{th}$ of the total. I understood the logic behind the total number of ...
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482 views

Presentation of Rubik's Cube group

The Rubik's Cube group is the group of permutations of the 20 cubes at the edges and vertices of a Rubik's group (taking into account their specific rotation) which are attainable by succesive ...
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Brute force method of solving the cube: How many moves would it take?

Given that Rubik's cube has finitely many positions, one possible "brute force" method to solve it would be to determine once a sequence of moves which eventually reaches every possible position of ...
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2answers
566 views

Orders of moves in the Rubik's cube

Playing with my Rubik's cube, I was thinking of facts about it that are immediate to mathematicians but novel to others. Here's one: Given a Rubik's cube in the starting position, any sequence of ...
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1answer
129 views

Rubik's cube notation [closed]

Various Rubik cube guides use the notation F and R to describe the front and right faces of the cube. However, in the illustrations they show the cube in isometric projection. So which side is right, ...
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3answers
2k views

Is there a “most random” state in Rubik's cube?

Is there a state in Rubik's cube which can be considered to have the highest degree of randomness (maximum entropy?) asssuming that the solved Rubik's cube has the lowest?
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221 views

Are actions in the $3\times 3\times 3$ rubik cube a group?

Are actions in the $3\times 3\times 3$ rubik cube a group? You can see here Rubik's Cube Not a Group? that $4\times 4\times 4$ rubik cubes or higher arent groups. But what about $3\times 3\times ...
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1answer
958 views

Rubik's Cube Not a Group?

I read online that although the 3x3x3 is a great example of a mathematical group, larger cubes aren't groups at all. How can that be true? There is obviously an identity and it is closed, so ...
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471 views

Rubik's cube puzzle

If we cut along the plane orthogonal to the largest diagonal of a Rubik's cube, what is the maximum number of small cubes can we cut? I thought this should be $9$, but apparently this is not the ...
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2answers
5k views

How to tell if a Rubik's cube is solvable

How can I determine if a certain Rubik's cube, that is in a certain state, is solveable? By "certain state" it could mean that the cube has been dismantled and put together again. And in my experience ...
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693 views

All possible permutations on a Rubik cube ($3\times3\times 3$) can be reached from the initial state?

If I were to represent a state in the Rubik cube as a permutation of the colors on the 9 tiles per side on all sides of the cube, could I reach all possible states (i.e. colorings) by the permutations ...
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1answer
316 views

Does solving a Rubik's cube imply alignment?

Today, I got my hands on a Rubik's cube with text on it. It looks like this: Now, I would love to know whether solving the right cube will also always correctly align the text on the cube or ...
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509 views

Mathematics Behind the 4×4 and 5×5 Rubik's Cube

A lot is known about the math behind the 3×3 Rubik's cube (symmetries, generators, group structure etc...). Is the same true for the 4×4 and 5×5 cubes? I haven't had much success finding this ...
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1answer
513 views

Solving a scrambled $3 \times 3 \times 3$ Rubik's Cube with at most 20 moves!

I read somewhere that any scrambled form of $3 \times 3 \times 3$ Rubik's cube can be solved using at most $20$ moves, and I just said "wow"! I am wondering can we prove this by mathematical ways? Or ...
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1answer
2k views

What are the 2125922464947725402112000 symmetries of a Rubik's Cube?

In a recent talk, Marcus du Sautoy says there are 2125922464947725402112000 (2.1*10^24) symmetries of a Rubik's cube, but doesn't explicitly identify what qualifies as a symmetry. What counts as a ...
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608 views

Rubik's cube interesting questions?

The upper bound for the number of moves required to solve a regular Rubik's cube has been shown to be 20. Two questions come to mind: Does this result have more general significance? What are the ...
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575 views

How many ways can I make six moves on a Rubik's cube?

I am writing a program to solve a Rubik's cube, and would like to know the answer to this question. There are 12 ways to make one move on a Rubik's cube. How many ways are there to make a sequence of ...
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2answers
302 views

How is done the calculation of the minimum number of movement to solve any configuration of Rubik's Cube?

I have read a few weeks ago that some mathematical researchers have discover that the minimum number of movements to solve any initial configuration of a Rubik's cube has been downsized to 20. How do ...
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1answer
561 views

How many disconnected graphs of the Rubik's cube exist?

Let us say that a Rubik's cube in a particular configuration is in a particular "state". All other configurations of this cube (other "states"), which can be achieved by rotations of the cube can be ...